Not sure about dip/pack and vet regulations for African animals but I have taken my salted/dried full-bodied sheep capes, skull caps and horns home with me from MX, Canada and Tajikistan as recently as 2018 and have helped several desert sheep clients do the same with their MX rams that I guided on Baja. For sheep and other CITES species, you need the CITES export permit from the hunted country. You can find the regulations for the USA side of things on the USFWS website. I printed the regs and took them with me. Definitely need a Form 3-177 as well.
You must enter the USA through a USFWS designated import Port of Entry such as LAX airport, Houston airport, Chicago airport or other ports of entry that are listed on the USFWS website list of approved ports of entry. Not every international US airport is on the list. You must declare that you have wildlife parts at US Customs and have the USFWS Officer and USDA Officer on duty do a quick inspection. The USFWS Officer will keep the original CITES permit so I always made a copy for my records before leaving MX, Canada or Tajikistan. Skins from MX were required to be frozen for 24 hours either before or after being salted/dried before traveling. Skins must be dried and salted enough so that they are “tack dried” (stiff, not wet). Skulls and horns have to be free of all flesh and soft tissue. I always called the USFWS Officer at LAX before my flight to make sure he was there when I landed. If my flight landed after normal business hours, there was an overtime fee.
I am not sure about the exact procedures for African animals. There might be Vet Certificates and dip/pack requirements that prevent you from taking freshly harvested animals home but I would bet that you could take animals harvested the year before home with you through an approved USFWS Port of Entry airport after doing some research.